For each Jacobian function, Table 22.4.1 gives its periods in the
z-plane in the left column, and the position of one of its poles in the second
row. The other poles are at congruent points,
which is the set of points
obtained by making translations by 2⁢m⁢K+2⁢n⁢i⁢K′, where m,n∈ℤ. For
example, the poles of sn⁡(z,k), abbreviated as sn in the
following tables, are at z=2⁢m⁢K+(2⁢n+1)⁢i⁢K′.

Three functions in the same column of Table 22.4.1 are
copolar, and four functions in the same row are coperiodic.

Table 22.4.2 displays the periods and zeros of the functions in the
z-plane in a similar manner to Table 22.4.1. Again, one member of
each congruent set of zeros appears in the second row; all others are generated
by translations of the form 2⁢m⁢K+2⁢n⁢i⁢K′, where m,n∈ℤ.

Figure 22.4.1 illustrates the locations in the z-plane of the
poles and zeros of the three principal Jacobian functions in the rectangle with vertices
0, 2⁢K, 2⁢K+2⁢i⁢K′, 2⁢i⁢K′. The other poles and zeros are at the congruent
points.

§22.4(ii) Graphical Interpretation via Glaisher’s Notation

Figure 22.4.2 depicts the fundamental unit cell in the
z-plane, with vertices s=0, c=K,
d=K+i⁢K′, n=i⁢K′. The set of points z=m⁢K+n⁢i⁢K′,
m,n∈ℤ, comprise the lattice
for the 12 Jacobian functions;
all other lattice unit cells are generated by translation of the fundamental unit
cell by m⁢K+n⁢i⁢K′, where again m,n∈ℤ.

Using the p,q notation of (22.2.10), Figure 22.4.2 serves
as a mnemonic for the poles, zeros, periods, and half-periods of the 12
Jacobian elliptic functions as follows. Let p,q be any two distinct letters
from the set s,c,d,n which appear in counterclockwise orientation at the
corners of all lattice unit cells. Then:
(a) In any lattice unit cell pq⁡(z,k) has a simple zero at z=p
and a simple pole at z=q.
(b) The difference between p and the nearest q is a half-period of
pq⁡(z,k). This half-period will be plus or minus a member of the triple
K,i⁢K′,K+i⁢K′; the other two members of this triple are
quarter periods of pq⁡(z,k).